Question

1. a chemist has 70 ml of a 50% methane solution. how much of a 80% solution must she add so the final solution is 60% methane? 5. pure acid (100%) is to be added to a 10% acid solution to obtain 54l of a 20% acid solution. what amounts of each should be used? 6. a coffee mix is to be made that sells for $2.50 by mixing two types of coffee. the cafe usually packages has 40 ml of guatemalan coffee for a price of $3.00. they want to blend it with the arabica coffee that costs $1.50. how much of the arabica should the cafe mix into the blend?

Explanation:

Problem 4

Step1: Set up the equation based on the amount of methane

Let $x$ be the volume (in mL) of the 80% methane - solution to be added. The amount of methane in the 50% solution is $0.5\times70$, the amount of methane in the 80% solution is $0.8x$, and the amount of methane in the final 60% solution is $0.6\times(70 + x)$. So the equation is $0.5\times70+0.8x=0.6\times(70 + x)$.
$35 + 0.8x=42+0.6x$

Step2: Solve the equation for $x$

Subtract $0.6x$ from both sides: $35 + 0.8x-0.6x=42+0.6x - 0.6x$, which gives $35 + 0.2x=42$.
Then subtract 35 from both sides: $35-35 + 0.2x=42 - 35$, so $0.2x=7$.
Divide both sides by 0.2: $x=\frac{7}{0.2}=35$.

Step1: Set up the equations

Let $x$ be the volume (in L) of pure acid (100% acid) and $y$ be the volume (in L) of the 10% acid - solution. We have two equations:
Equation 1 (total volume): $x + y=54$.
Equation 2 (amount of acid): $1.0x+0.1y=0.2\times54$.
From the first equation, $x = 54 - y$. Substitute $x = 54 - y$ into the second equation: $1\times(54 - y)+0.1y = 10.8$.
$54-y + 0.1y=10.8$.

Step2: Solve the equation for $y$

Combine like - terms: $54-(1 - 0.1)y=10.8$, so $54-0.9y=10.8$.
Subtract 54 from both sides: $-0.9y=10.8 - 54=-43.2$.
Divide both sides by $-0.9$: $y=\frac{-43.2}{-0.9}=48$.
Then find $x$ using $x = 54 - y$, so $x=54 - 48 = 6$.

Step1: Set up the equations

Let $x$ be the volume (in mL) of Guatemalan coffee and $y$ be the volume (in mL) of Arabica coffee. We know $x = 40$.
The cost - based equation is $3x+1.5y=2.5(x + y)$.
Substitute $x = 40$ into the equation: $3\times40+1.5y=2.5\times(40 + y)$.
$120+1.5y=100 + 2.5y$.

Step2: Solve the equation for $y$

Subtract $1.5y$ from both sides: $120+1.5y-1.5y=100 + 2.5y-1.5y$, which gives $120=100 + y$.
Subtract 100 from both sides: $y=120 - 100 = 20$.

Answer:

35 mL

Problem 5